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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without Derivatives</a>
</h2></div></div></div>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket and
      Solve Root</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/TOMS748.html">Algorithm TOMS 748:
      Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination
      Condition Functors</a></span></dt>
<dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt>
</dl></div>
<h5>
<a name="math_toolkit.roots_noderiv.h0"></a>
      <span class="phrase"><a name="math_toolkit.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.synopsis">Synopsis</a>
    </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"> <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
 <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span>
 <span class="comment">// Bisection</span>
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">bisect</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">bisect</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">bisect</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
       <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

 <span class="comment">// Bracket and Solve Root</span>
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
       <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
       <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
       <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="comment">// TOMS 748 algorithm</span>
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">toms748_solve</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">toms748_solve</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
       <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">toms748_solve</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
    <span class="identifier">toms748_solve</span><span class="special">(</span>
       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fa</span><span class="special">,</span>
       <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">fb</span><span class="special">,</span>
       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
       <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
       <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

 <span class="comment">// Termination conditions:</span>
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>

 <span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
 <span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
 <span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>

 <span class="special">}}}</span> <span class="comment">// boost::math::tools namespaces</span>
</pre>
<h5>
<a name="math_toolkit.roots_noderiv.h1"></a>
      <span class="phrase"><a name="math_toolkit.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.description">Description</a>
    </h5>
<p>
      These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span> -
      <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
    </p>
<p>
      The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
      functions use <a class="link" href="roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a>
      by Alefeld, Potra and Shi that is asymptotically the most efficient known,
      and has been shown to be optimal for a certain classes of smooth functions.
      Variants with and without <a class="link" href="../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policies</a> are provided.
    </p>
<p>
      Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
      is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
      routine which can be useful in its own right in some situations, or alternatively
      for narrowing down the range containing the root, prior to calling a more advanced
      algorithm.
    </p>
<p>
      All the algorithms in this section reduce the diameter of the enclosing interval
      with the same asymptotic efficiency with which they locate the root. This is
      in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
      significantly reduce the enclosing interval, even though they rapidly approach
      the root. This is also in contrast to some other derivative-free methods (for
      example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a>
      which only reduces the enclosing interval on the final step. Therefore these
      methods return a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> containing the enclosing interval found,
      and accept a function object specifying the termination condition.
    </p>
<p>
      Three function objects are provided for ready-made termination conditions:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          <span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative
          error in the enclosing interval is below a certain threshold.
        </li>
<li class="listitem">
          <span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span> are
          useful for certain statistical applications where the result is known to
          be an integer.
        </li>
<li class="listitem">
          Other user-defined termination conditions are likely to be used only rarely,
          but may be useful in some specific circumstances.
        </li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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